The idea, step by step
A shape carried through space
This chapter uses right prisms and right circular cylinders, whose matching ends lie directly opposite one another. A prism has two matching parallel ends. Slices parallel to those ends have the same shape and area at every point along its length. A triangular prism has triangular cross-sections; a rectangular prism has rectangular cross-sections.
The term cross-section describes the flat region made by cutting a solid with a plane. The cutting direction matters. A cut across a cylinder parallel to its circular base gives a circle, but a lengthwise cut through its axis gives a rectangle.
Why cross-sections matter
Imagine building a prism from a stack of equal thin slices. The area of one slice measures how much space one layer covers. Multiplying that area by the prism’s perpendicular length gives its volume.
Use the cross-section perpendicular to the prism’s length when applying this volume rule. A pyramid tapers, so its parallel slices change size; the prism rule cannot be used unchanged.
A common trap
Calling a long side face the cross-section used in the volume formula.
Instead: Choose the repeated end-shaped slice perpendicular to the prism’s length.
See the thinking
Choose the right shape
A solid has two matching triangular ends and rectangular side faces. Name a cross-section parallel to an end.
- The matching ends identify a triangular prism.
- A cut parallel to either triangular end has the same triangular shape and dimensions.
Calculate a cross-sectional area
A prism’s cross-section is a triangle with base 6 cm and perpendicular height 4 cm. Find its cross-sectional area.
- Use the triangle formula, not the length of the solid.
- cm².
- The answer is an area, so the unit is cm².
Check your understanding
A rectangular prism has an end face 3 cm by 5 cm. What is the area of a parallel cross-section?
Show answer
15 cm².
Each parallel cross-section matches the end: cm².
Your turn
6 questions · 11 marks · Work at your own pace.
Read the questions online
- Name a cross-section parallel to an end of a pentagonal prism.1 mark
- Name a cross-section of a cylinder cut parallel to its circular base.1 mark
- A prism has a rectangular end 7 cm by 4 cm. Find the area of a parallel cross-section.2 marks
- A triangular prism has end base 8 m and end perpendicular height 3 m. Find the end area.2 marks
- A student applies cross-sectional area × length to a pyramid. Explain the problem.3 marks
- A solid has a circular base and tapers to one point. Is it a prism or a cylinder? Explain.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Name a cross-section parallel to an end of a pentagonal prism.1 mark
A prism repeats its end shape along its length.
A pentagon congruent to the end. - Name a cross-section of a cylinder cut parallel to its circular base.1 mark
The cut has the same orientation as a circular end.
A circle. - A prism has a rectangular end 7 cm by 4 cm. Find the area of a parallel cross-section.2 marks
cm².
28 cm². - A triangular prism has end base 8 m and end perpendicular height 3 m. Find the end area.2 marks
m².
12 m². - A student applies cross-sectional area × length to a pyramid. Explain the problem.3 marks
The prism rule relies on a constant cross-section.
A pyramid narrows towards its vertex.
A pyramid’s parallel cross-sections change size, so one fixed end area cannot be used throughout. - A solid has a circular base and tapers to one point. Is it a prism or a cylinder? Explain.2 marks
Its cross-sections parallel to the base become smaller.
A prism or cylinder has a constant cross-section.
Neither; it is a cone.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum
Circumference and area of circles; perimeter/area of composite shapes; choosing and converting area units; cross-sections of prisms; prism volume and capacity; mixed measurement applications.
Official curriculum source ↗